Deflection In Cantilever Beam Calculator

Maximum End Deflection Formula:

\[ \delta_{max} = \frac{P L^3}{3 E I} \]

Load (P):

Length (L):

Modulus of Elasticity (E):

Moment of Inertia (I):

m⁴

Maximum Deflection (δ_max):

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1. What is Maximum End Deflection?

Maximum end deflection (δ_max) is the largest displacement that occurs at the free end of a cantilever beam when a concentrated load is applied at that end. It's a critical parameter in structural engineering for ensuring beams meet design requirements.

2. How Does the Calculator Work?

The calculator uses the cantilever beam deflection formula:

\[ \delta_{max} = \frac{P L^3}{3 E I} \]

Where:

\( P \) — Concentrated load at the free end (N)
\( L \) — Length of the beam (m)
\( E \) — Modulus of elasticity of the material (Pa)
\( I \) — Moment of inertia of the cross-section (m⁴)

Explanation: The formula calculates the maximum vertical displacement at the free end of a cantilever beam subjected to a point load at its end.

3. Importance of Deflection Calculation

Details: Calculating deflection is crucial for structural design to ensure that beams don't deflect excessively under load, which could lead to serviceability issues or structural failure.

4. Using the Calculator

Tips: Enter all values in consistent SI units. Load (P) in Newtons, length (L) in meters, modulus of elasticity (E) in Pascals, and moment of inertia (I) in meters to the fourth power.

5. Frequently Asked Questions (FAQ)

Q1: What is a cantilever beam?
A: A cantilever beam is a rigid structural element supported at only one end, with the other end free. It's commonly used in construction, aerospace, and mechanical applications.

Q2: What are typical deflection limits?
A: Deflection limits vary by application but are typically L/240 to L/360 for live loads, where L is the span length, to prevent visible sagging or damage to attached elements.

Q3: Does this formula work for distributed loads?
A: No, this specific formula is for a concentrated load at the free end. Different formulas exist for uniformly distributed loads or loads applied at other points.

Q4: What affects moment of inertia (I)?
A: Moment of inertia depends on the cross-sectional shape and dimensions. It represents the beam's resistance to bending, with higher values indicating stiffer sections.

Q5: How does material choice affect deflection?
A: Materials with higher modulus of elasticity (E) values (like steel vs. wood) will deflect less under the same load, making them stiffer choices for cantilever applications.